Padé approximants for finite time ruin probabilities

In this article, we investigate Padé approximants of hyper-exponential to base on the first moments and the matrix-exponential representation of Padé approximated function. An explicit formula is given for the Laplace transform in the time to calculate finite time ruin probabilities of classical Cramer–Lundberg model. This formula generalizes the ultimate ruin probabilities formula of Asmussen and Rolski (1991) [12]. To illustrate this formula, several numerical examples with different values uu are given.